Investigating the effects of the fixed and varying dispersion parameters of Poisson-gamma models on empirical Bayes estimates.

Traditionally, transportation safety analysts have used the empirical Bayes (EB) method to improve the estimate of the long-term mean of individual sites; to correct for the regression-to-the-mean (RTM) bias in before-after studies; and to identify hotspots or high risk locations. The EB method combines two different sources of information: (1) the expected number of crashes estimated via crash prediction models, and (2) the observed number of crashes at individual sites. Crash prediction models have traditionally been estimated using a negative binomial (NB) (or Poisson-gamma) modeling framework due to the over-dispersion commonly found in crash data. A weight factor is used to assign the relative influence of each source of information on the EB estimate. This factor is estimated using the mean and variance functions of the NB model. With recent trends that illustrated the dispersion parameter to be dependent upon the covariates of NB models, especially for traffic flow-only models, as well as varying as a function of different time-periods, there is a need to determine how these models may affect EB estimates. The objectives of this study are to examine how commonly used functional forms as well as fixed and time-varying dispersion parameters affect the EB estimates. To accomplish the study objectives, several traffic flow-only crash prediction models were estimated using a sample of rural three-legged intersections located in California. Two types of aggregated and time-specific models were produced: (1) the traditional NB model with a fixed dispersion parameter and (2) the generalized NB model (GNB) with a time-varying dispersion parameter, which is also dependent upon the covariates of the model. Several statistical methods were used to compare the fitting performance of the various functional forms. The results of the study show that the selection of the functional form of NB models has an important effect on EB estimates both in terms of estimated values, weight factors, and dispersion parameters. Time-specific models with a varying dispersion parameter provide better statistical performance in terms of goodness-of-fit (GOF) than aggregated multi-year models. Furthermore, the identification of hazardous sites, using the EB method, can be significantly affected when a GNB model with a time-varying dispersion parameter is used. Thus, erroneously selecting a functional form may lead to select the wrong sites for treatment. The study concludes that transportation safety analysts should not automatically use an existing functional form for modeling motor vehicle crashes without conducting rigorous analyses to estimate the most appropriate functional form linking crashes with traffic flow.